What is the meaning of the last argument passed to the ColorFunction of a ParametricPlot3D?

Question

According to the docs, the meaning of the arguments passed to ParametricPlot3D is $x, y, z, u, v$. This is 5 arguments: the Cartesian coordinates and the two parameters. Similarly, for ParametricPlot it's $x, y, u, v$.

Let's look at how many arguments are actually passed to the ColorFunction:

Reap[ParametricPlot3D[{Cos[u] Cos[v], Sin[u] Cos[v], Sin[v]}, 
       {u, 0, 2 Pi}, {v, -Pi/2, Pi/2}, ColorFunction -> ((Sow[{##}]; Gray) &)]]

If you evaluate this, you'll notice that actually 6 arguments are passed to the colour function, but the last one is always zero. If we use the one-parameter form of ParametricPlot3D, then still six arguments will be passed, but now the last two are zero.

Reap[ParametricPlot3D[{Cos[u], Sin[u], 0}, {u, 0, 2 Pi}, 
  ColorFunction -> ((Sow[{##}]; Gray) &)]]

Similarly, the colour function of ParametricPlot receives 5 arguments instead of 4.

What is the meaning of the last argument passed to the ColorFunction of these functions? Does it have any use? Does ParametricPlot3D have a form where the last argument of the colour function will not be zero and has an application?

Answer 1

This is a bit speculative but I note that SphericalPlot3D[] supplies its ColorFunction with 6 args, being x, y, z, theta, phi, and r. SphericalPlot3D can be shown to call ParametricPlot3D, so I'd surmise the extra argument is left there for compatibility.